TOKYO—Toshiba Corp. is lagging behind rival chip makers in securing next year’s supply of silicon wafers, people in the industry said, showing how the company’s troubles are hamstringing operations at its principal profit center.

Toshiba is trying to sell its memory-chip unit for some $20 billion in a bid to survive after its U.S. nuclear unit, Westinghouse Electric, filed for bankruptcy protection in March and left Toshiba with multibillion-dollar losses. But the sale process has gotten bogged down. At a meeting...